On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed.